The Shanks–Rényi prime number race with many contestants

Under certain plausible assumptions, M. Rubinstein and P.Sarnak solved the Shanks–Rényi race problem by showingthat the set of real numbers $x\geq 2$ such that$\pi(x;q,a_1)>\pi(x;q,a_2)>\cdots>\pi(x;q,a_r)$ has apositive logarithmic density $\delta_{q;a_1,\dots,a_r}$.Furthermore, they established that if $r$ is fixed,$\delta_{q;a_1,\dots,a_r}\to 1/r!$ as $q\to \infty$. Inthis paper, we investigate the size of these densities whenthe number of contestants $r$ tends to infinity with $q$.In particular, we deduce a strong form of a recentconjecture of Feuerverger and Martin which states that$\delta_{q;a_1,\dots,a_r}=o(1)$ in this case. Among ourresults, we prove that $\delta_{q;a_1,\dots,a_r}\sim 1/r!$in the region $r=o(\sqrt{\log q})$ as $q\to\infty$. We alsobound the order of magnitude of these densities beyond thisrange of $r$. For example, we show that when $\log q\leqr\leq \phi(q)$, $\delta_{q;a_1,\dots,a_r}\ll_{\epsilon}q^{-1+\epsilon}$.

Keywords

Shanks–Rényi race problem, primes in arithmetic progressions, zeros of Dirichlet L-functions

2010 Mathematics Subject Classification

Primary 11N13. Secondary 11M26, 11N69.

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Published 8 November 2012